Theorem Goal

In this paper,we focus all our attention on this frational space and more pricisely the fractional space,the Bessel potential space $H ^ { s , p } ( \Omega )$ .Before we start any discussion,we give the universal definition of the Bessel potential space .

$\left. \begin{array}{l}{ H ^ { s , p } ( R ^ { N } ) = \{ u \in L ^ { p } ( R ^ { N } ) : ( - \Delta ) ^ { s / 2 } u \in L ^ { p } ( R ^ { N } ) \} }\\{ \tilde { H } ^ { s , p } ( \Omega ) = \{ u \in H ^ { s , p } ( R ^ { N } ) : u \equiv 0 \quad \text { on } R ^ { N } \backslash \Omega \} }\end{array} \right.$

$\begin{aligned}H ^ { s , p } ( R ^ { N } ) &= \{ u \in L ^ { p } ( R ^ { N } ) : ( - \Delta ) ^ { s / 2 } u \in L ^ { p } ( R ^ { N } ) \} \\ &H ^ { s , p } ( R ^ { N } ) &=u \in H ^ { s , p } ( R ^ { N } ) : u \equiv 0 \quad \text { on } R ^ { N\backslash\Omega}\}end{aligned}$

Theorem. Let $u \subset H ^ { s , p } ( \Omega )$ which is the Bessel potential space and to some degree ,equivalent to the fractional Sobolev spaces $W ^ { s , p }( \Omega )$ where both of them in bounded domain $( \Omega )$ , be a sequence of functions such that $\| ( - \Delta ) ^ { s } u \| _ { L^2 ( \Omega ) } \leq 1$ and $u_{\epsilon} \rightharpoonup u_{0}$ weakly in $H_{0}^{s}(\Omega)$ . Then for any $p < 1 / ( 1 - \| ( - \Delta ) ^ { s /2} u _ { \epsilon} \| _ { 2 } ^ { 2 } )$ .

$\underset { \epsilon \rightarrow 0 } { \operatorname { lim sup } } \int _ { \Omega } e ^ { {\alpha _ { n , 2 }} p u _ { \epsilon } ^ { 2 } } d x < + \infty$

$W ^ { s , p } ( \Omega ) : = \{ u \in L ^ { p } ( \Omega ) : \frac { | u ( x ) - u ( y ) | } { | x - y | ^ { \frac { n } { p } + s } } \in L ^ { p } ( \Omega \times \Omega ) \}$

$H ^ { s , p } ( \Omega )$ $W ^ { s , p }$